HANDS-ON RELAY SCHOOL 2019 Steve Laslo System Protection and Control Specialist Bonneville Power Administration Presentation Information This presentation is a revision to Phasor Diagrams, presented at the Hands-On Relay School for many years by Ron Alexander. Much of the content in this presentation is based on J. Lewis Blackburn’s fantastic reference: Protective Relaying Principles and Applications (multiple editions). Our Primary Objective for this presentation is to enhance attendee knowledge by: Reviewing Phasor fundamentals Examining In-Service/Load checks Our Secondary Objective (time permitting) will add: Reviewing Fault Phasors Using phasors to determine phase shift across three-phase transformer banks Phasors – Why? • Tool for understanding the power system during load and fault conditions. • Assists a person in understanding principles of relay operation for testing and analysis of relay operations. • Allows technicians to simulate faults that can be used to test relays. • Common language of power protection engineers and technicians. • Provides both mathematical and graphical view of System conditions. Phasor Definitions A line used to represent a complex electrical quantity as a vector. (Google) A rotating vector representing a quantity, such as an alternating current or voltage, that varies sinusoidally. (Collins Dictionary) A vector that represents a sinusoidally varying quantity, as a current or voltage, by means of a line rotating about a point in a plane, the magnitude of the quantity being proportional to the length of the line and the phase of the quantity being equal to the angle between the line and a reference line. (Dictionary.com) Phasor Representation Phasor Rotation Here we can see a plot of an electrical quantity and its phasor representation. Note the phasor has a constant (usually RMS) magnitude that rotates while the actual electrical quantity varies sinusoidally over time. Multiple Phasors Here we can see two phasors of the same frequency rotating. Note that their phase relationship to each other is constant – in this case, the blue phasor leads the red phasor by some constant angle that does not change. PLOTTING PHASORS * V P I CARTESIAN COORDINATE SYSTEM Phasor Representation Consider an example phasor ‘c’ having a magnitude of 120VRMS and a phase angle of 30 degrees: Rectangular Form: c = x + jy c = 104 + j60 V Polar Form: c = |c|∠θ c = 120∠30° V Complex Form: c = |c|(cosθ + jsinθ) c = 120(cos(30) + jsin(30)) V Exponential Form: c = |c|ejθ c = 120ej30 V Phasor Conversion Rectangular <> Polar Conversion: Rectangular Form: c = x + jy c = 104 + j60 Polar Form: c = c∠θ c = 120∠30° Trig Functions (right triangles only): Sin(θ) = opposite / hypotenuse Cos(θ) = adjacent / hypotenuse Tan(θ) = opposite / adjacent Pythagorean Theorem c2 = a2 + b2 Phasor Conversion Rectangular to Polar Conversion: Rectangular Form: c = x + jy c = 104 + j60 c2 = a2 + b2 c2 = 1042 + 602 >> c = 120 Tan (θ) = opposite / adjacent Tan (θ) = 60 / 104 >> θ = 30° Converted: c = 120∠30° Phasor Conversion Polar to Rectangular Conversion: Polar Form: c = |c|∠θ c = 120∠30° Sin(θ) = o/h o = h* Sin(θ) o = 120 * Sin(30) = 60 Cos(θ) = a/h a = h * Cos(θ) a = 120 * Cos(30) = 104 Converted: c = 104 + j60 Operators Two ‘Operators’ related to phasors are commonly used in the Power world: ‘j’ and ‘a’ Mathematically ‘j’ is an imaginary number representing the imaginary (reactive) portion of a phasor: j = √-1 Graphically, it is a ‘rotator’ constant with an angle of 90° It can also be viewed as a ‘unit phasor’ always having a value of 1∠90° The ‘a’ Operator, commonly used when working with Symmetrical Components. Graphically it is a ‘rotator’ constant with an angle of 120° It is also a ‘unit phasor’ with a value of 1∠120° Combining Phasors Two common operations are performed with phasors: Adding / Subtracting Multiplying / Dividing It’s generally easier to add/subtract in rectangular form and easier to multiply/divide in polar form. When adding/subtracting in rectangular form, add/subtract the real and reactive components (respectively): Example: (2+j3) + (3+j4) = (5+j7) When multiplying/dividing in polar form, multiply/divide the magnitude, and add/subtract the angle: Example: (10∠30°) * (5∠45°) = 50∠75°) Adding Phasors Graphically You can also add vectors graphically by connecting them head to tail. The resultant is the phasor originating at the origin of the first arrow and ending at the head of the last arrow. Here, if we have the following: Va = 120∠0°V Vb = 120∠-120°V Vc = 120∠120°V Va + Vb + Vc = 0 Adding Phasors Graphically In this example, the VAB voltage is VA-VB. We subtract VB by reversing the VB phasor and adding it to VA. VAB = VA-VB = VA+(-VB) Sinusoidal Waveforms Note that adding or subtracting sinusoidal waveforms simply produces a resultant sinusoidal waveform of the same frequency. 150 100 50 VA VB 0 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 570 600 630 660 690 720 VAB -VB -50 -100 -150 Which One is Easier? 150 100 50 VA VB 0 0 30 60 90 120 150 180 210 240 270 300 330 360 390 420 450 480 510 540 570 600 630 660 690 720 VAB -VB -50 -100 -150 Phasors – Conjugation Most commonly used in power calculation: P=EI where the conjugate of I is sometimes used and shown as: P=EI* Conjugates: c = x - jy c = c∠-Φ c = |c|(cosΦ - jsinΦ) c = |c|e-jΦ Phasors – Conjugation Calculating: P = E * I we get: P = 120∠0°V * 0.42 ∠-45°A = 50.9∠-45°VA Placing this phasor on one of our previous Power Flow diagrams gives →→→ But Blackburn and many others use +Q in the 1st Quadrant. If we calculate P using the conjugate of the current: P = E * I* P = 120∠0°V * 0.42 ∠+45°A = 50.9∠+45°VA This places P in the 1st Quadrant. Phasors – Conjugation It’s extremely important to remember that phasor diagrams are descriptors of circuit information. How they are placed on a diagram does not change the electrical qualities of a circuit. Phase Sequence Phase Sequence is the order in which phasors pass a reference point. In Symmetrical Components, you will hear the terms: positive sequence, negative sequence, and zero sequence. Positive Sequence = ABC Negative Sequence = ACB. Zero sequence is all 3 phases rotating together at the same angle. Resistor AC Response Unity Power Factor Waveform 60 50 40 30 Magnitude 20 10 0 0 30 60 90 120 150 180 210 240 270 300 330 360 -10 -20 Angle in Degrees Voltage Current Power Phasor Diagram Inductor AC Response 90 Degree Lagging Waveform 25 20 15 10 5 0 -5 0 30 60 90 120 150 180 210 240 270 300 330 360 Magnitude -10 -15 -20 -25 Angle in Degrees Voltage Current Power Phasor Diagram Capacitor AC Response 90 Degree Leading Waveform 30 20 10 0 0 30 60 90 120 150 180 210 240 270 300 330 360 Magnitude -10 -20 -30 Angle in Degrees Voltage Current Power Phasor Diagram Phasor Nomenclature Phasor Nomenclature Phasor Nomenclature Phasor diagrams can be drawn ‘open’ or ‘closed’. Polarity Current into the primary polarity and out of the secondary polarity are (essentially) in-phase. Voltage drop from polarity to non-polarity on both windings are (essentially) in-phase. Polarity and Phasors When dealing with Phasors, we make assumptions about current and voltage polarity on our circuit diagram so polarity is also extremely important. Consider the following circuit: we’ll do ‘in- service’ checks of the relays in the bottom Chemawa current circuit. Substation Phasors in Practice Here we’ve zoomed in on the current circuit we are interested in: We’ll assume for convenience that the normal load flow for this circuit is from the Main Bus out into the line – the ‘up’ direction on this print. Phasors in Practice Now we’ll plan on where we will take secondary current readings. The test switches are made for this so we’ll use terminals 11/12, 15/16, and 19/20 on test switch 3L. Phasors in Practice We’ll also plan on checking the voltages to the relays for proper magnitude and polarity. We’ll use the same 3L test switch, terminals 1-8. Phasors in Practice Which of the following phasor diagrams is correct for the circuit we are doing in-service on? It is indeterminate / uncertain; both could be correct, or wrong… Why? We haven’t completed our circuit diagram (which we know should accompany our phasor diagram). Phasors in Practice Since we already assumed a load flow direction, now we just need to show on our circuit diagram our assumed polarity at our measurement point. This determines how we ‘jack in’ to the current circuit using a phase-angle meter. In this diagram we will insert test equipment with assumed current ‘in’ on terminals 11, 15, and 19. IC IB Assumed Direction IA of Load Flow Phasors in Practice Let’s show/document our measurement points for the voltages too. VAN = (3L) 1-7, VBN = (3L) 3-7, VCN = (3L) 5-7 VA VB VC VN In-Service Sheet Here is a in-service sheet based on our work and our IC measurements VCN based on our circuit diagram. VAN What does it tell IB us? IA VBN It tells us our ‘assumed’ current direction was correct, and load is flowing from Chemawa to 173 100 Santiam. It also tells us the load at Santiam is moderately Inductive. Phasors in Practice Revisiting this slide, it appears Phasor Diagram #1 is correct – at least when it is paired with the circuit diagrams we’ve previously looked at.